vacuum fluctuations in domains with moving …vacuum fluctuations and the equivalence principle ce...

ICE

Vacuum Fluctuations inDomains with Moving Boundaries

and the Dark Energy Issue

EMILIO ELIZALDE

ICE/CSIC & IEEC, UAB, Barcelona

4th Dark Side of the Universe, BUE El Cairo, 1-5 June, 2008

4th Dark Side of the Universe, BUE El Cairo, 1-5 June, 2008 – p. 1/19

Outline of this presentation

The Casimir Effect: Theory, Experiments, Uses




Fulling-Davies Theory (Dynamical CE)





Vacuum Fluctuations and the Equivalence Principle






CE and Accelerated Expansion (Dark Energy)






CE and Accelerated Expansion (Dark Energy)

Gravity Equations as Equations of State


Zero point energy

QFT vacuum to vacuum transition: 〈0|H|0〉


Zero point energy


Spectrum, normal ordering (harm oscill):

H =

(n +

1

2

)λn an a†n


Zero point energy



H =

(n +

1

2

)λn an a†n

〈0|H|0〉 =~ c

2

∑

n

λn =1

2tr H


Zero point energy



H =

(n +

1

2

)λn an a†n

〈0|H|0〉 =~ c

2

∑

n

λn =1

2tr H

gives ∞ physical meaning?


Zero point energy



H =

(n +

1

2

)λn an a†n

〈0|H|0〉 =~ c

2

∑

n

λn =1

2tr H


Regularization + Renormalization ( cut-off, dim, ζ )


Zero point energy



H =

(n +

1

2

)λn an a†n

〈0|H|0〉 =~ c

2

∑

n

λn =1

2tr H


Regularization + Renormalization ( cut-off, dim, ζ )

Even then: Has the final value real sense ?


The Casimir Effect

vacuum

BC

F

Casimir Effect


The Casimir Effect

vacuum

BC

F

Casimir Effect

BC e.g. periodic


The Casimir Effect

vacuum

BC

F

Casimir Effect

BC e.g. periodic=⇒ all kind of fields


The Casimir Effect

vacuum

BC

F

Casimir Effect

BC e.g. periodic=⇒ all kind of fields=⇒ curvature or topology


The Casimir Effect

vacuum

BC

F

Casimir Effect


Universal process:


The Casimir Effect

vacuum

BC

F

Casimir Effect


Universal process:

Sonoluminiscence (Schwinger)

Cond. matter (wetting 3He alc.)

Optical cavities

Direct experim. confirmation


The Casimir Effect

vacuum

BC

F

Casimir Effect


Universal process:



Optical cavities


Van der Waals, Lifschitz theory


The Casimir Effect

vacuum

BC

F

Casimir Effect


Universal process:



Optical cavities


Van der Waals, Lifschitz theoryDynamical CE ⇐

Lateral CE

Extract energy from vacuum

CE and the cosmological constant ⇐


The standard approach


The standard approach=⇒ Casimir force: calculatedby computing change in zeropoint energy of the em field



=⇒ But Casimireffects can be calculatedas S-matrix elements:Feynman diagrs with ext. lines




In modern language the Casimir energy can be expressed in terms of thetrace of the Greens function for the fluctuating field in the background ofinterest (conducting plates)

E =~

2πIm

∫dωω Tr

∫d3x [G(x, x, ω + iǫ) − G0(x, x, ω + iǫ)]




In modern language the Casimir energy can be expressed in terms of thetrace of the Greens function for the fluctuating field in the background ofinterest (conducting plates)

E =~

2πIm

∫dωω Tr

∫d3x [G(x, x, ω + iǫ) − G0(x, x, ω + iǫ)]

G full Greens function for the fluctuating fieldG0 free Greens function Trace is over spin


EC = 〈〉plates − 〈〉no plates



1

πIm

∫[G(x, x, ω + iǫ) − G0(x, x, ω + iǫ)] =

d∆N

dω

change in the density of states due to the background



1

πIm

∫[G(x, x, ω + iǫ) − G0(x, x, ω + iǫ)] =

d∆N

dω


=⇒ A restatement of the Casimir sum over shifts in zero-point energies

~

2

∑(ω − ω0)



1

πIm

∫[G(x, x, ω + iǫ) − G0(x, x, ω + iǫ)] =

d∆N

dω



~

2

∑(ω − ω0)

=⇒ Lippman-Schwinger eq. allows full Greens f, G, be expanded as aseries in free Green’s f, G0, and the coupling to the external field



1

πIm

∫[G(x, x, ω + iǫ) − G0(x, x, ω + iǫ)] =

d∆N

dω



~

2

∑(ω − ω0)

=⇒ Lippman-Schwinger eq. allows full Greens f, G, be expanded as aseries in free Green’s f, G0, and the coupling to the external field

=⇒ “Experimental confirmation of the Casimir effect does not establish thereality of zero point fluctuations” [R. Jaffe et. al.]


The Dynamical Casimir EffectS.A. Fulling & P.C.W. Davies, Proc Roy Soc A348 (1976)



Moving mirrors modify structure of quantum vacuum

Creation and annihilation of photons; once mirrors return to rest,some produced photons may still remain: flux of radiated particles





For a single, perfectly reflecting mirror:# photons & energy diverge while mirror moves






Several renormalization prescriptions have been usedin order to obtain a well-defined energy







Problem: for some trajectories this finite energy is not a positivequantity and cannot be identified with the energy of the photons







Problem: for some trajectories this finite energy is not a positivequantity and cannot be identified with the energy of the photons

Moore; Razavy, Terning; Johnston, Sarkar; Dodonov et al;Plunien et al; Barton, Eberlein, Calogeracos; Ford, Vilenkin;Jaeckel, Reynaud, Lambrecht; Brevik, Milton et al;Dalvit, Maia-Neto et al; Law; Parentani, ...


A CONSISTENT APPROACH:J. Haro & E.E., PRL 97 (2006); arXiv:0705.0597



Partially transmitting mirrors, which become transparent tovery high frequencies (analytic matrix)

Proper use of a Hamiltonian method & corresponding renormalization





Proved both: # of created particles is finite & their energy is alwayspositive, for the whole trajectory during the mirrors’ displacement






The radiation-reaction force acting on the mirrors owing to emission-absorption of particles is related with the field’s energy through theenergy conservation law: energy of the field at any t equals (withopposite sign) the work performed by the reaction force up to time t







Such force is split into two parts: a dissipative forcewhose work equals minus the energy of the particles that remain& a reactive force vanishing when the mirrors return to rest







Such force is split into two parts: a dissipative forcewhose work equals minus the energy of the particles that remain& a reactive force vanishing when the mirrors return to rest

The dissipative part we obtain agrees with the other methods.But those have problems with the reactive part, which in generalyields a non-positive energy =⇒ EXPERIMENT


SOME DETAILS OF THE METHOD

Hamiltonian method for neutral Klein-Gordon field in a cavity Ωt, withboundaries moving at a certain speed v << c, ǫ = v/c

(of order 10−8 in Kim, Brownell, Onofrio, PRL 96 (2006) 200402)





Assume boundary at rest for time t ≤ 0 and returns to its initialposition at time T






Hamiltonian density conveniently obtained using the method inJohnston, Sarkar, JPA 29 (1996) 1741







Lagrangian density of the field

L(t,x) =1

2

[(∂tφ)2 − |∇xφ|2

], ∀x ∈ Ωt ⊂ R

n, ∀t ∈ R







Lagrangian density of the field

L(t,x) =1

2

[(∂tφ)2 − |∇xφ|2

], ∀x ∈ Ωt ⊂ R

n, ∀t ∈ R

Hamiltonian. Transform moving boundary into fixed one by(non-conformal) change of coordinates

R : (t,y) → (t(t,y),x(t,y)) = (t,R(t,y))

transform Ωt into a fixed domain Ω

Ω: (t(t,y),x(t,y)) = R(t,y) = (t,R(t,y))

(with t the new time)4th Dark Side of the Universe, BUE El Cairo, 1-5 June, 2008 – p. 9/19

CASE OF A SINGLE, PARTIALLY TRANSMITTING MIRROR



Seminal Davis-Fulling model [PRSL A348 (1976) 393]renormalized energy negative while the mirror moves:cannot be considered as the energy of the produced particles at time t

[cf. paragraph after Eq. (4.5)]




[cf. paragraph after Eq. (4.5)]Our interpretation: a perfectly reflecting mirror is non-physical.Consider, instead, a partially transmitting mirror, transparent to highfrequencies (math. implementation of a physical plate).




[cf. paragraph after Eq. (4.5)]Our interpretation: a perfectly reflecting mirror is non-physical.Consider, instead, a partially transmitting mirror, transparent to highfrequencies (math. implementation of a physical plate).Trajectory (t, ǫg(t)). When mirror at rest, scattering described by matrix

S(ω) =

s(ω) r(ω) e−2iωL

r(ω) e2iωL s(ω)

=⇒ S matrix is taken to be: (x = L position of the mirror)




[cf. paragraph after Eq. (4.5)]Our interpretation: a perfectly reflecting mirror is non-physical.Consider, instead, a partially transmitting mirror, transparent to highfrequencies (math. implementation of a physical plate).Trajectory (t, ǫg(t)). When mirror at rest, scattering described by matrix

S(ω) =

s(ω) r(ω) e−2iωL

r(ω) e2iωL s(ω)

=⇒ S matrix is taken to be: (x = L position of the mirror)

→ Real in the temporal domain: S(−ω) = S∗(ω)

→ Causal: S(ω) is analytic for Im (ω) > 0

→ Unitary: S(ω)S†(ω) = Id→ The identity at high frequencies: S(ω) → Id, when |ω| → ∞

s(ω) and r(ω) meromorphic (cut-off) functions(material’s permitivity and resistivity)


RESULTS ARE REWARDING:



In our Hamiltonian approach

〈FHa(t)〉 = −ǫ

2π2

∫∞

0

∫∞

0

dωdω′ωω′

ω + ω′Re

[e−i(ω+ω

′)t gθt(ω + ω′)]

×[|r(ω) + r∗(ω′)|2 + |s(ω) − s∗(ω′)|2] + O(ǫ2)

Note this integral diverges for a perfect mirror (r ≡ −1, s ≡ 0,ideal case), but nicely converges for our partially transmitting(physical) one where r(ω) → 0, s(ω) → 1, as ω → ∞




〈FHa(t)〉 = −ǫ

2π2

∫∞

0

∫∞

0

dωdω′ωω′

ω + ω′Re

[e−i(ω+ω

′)t gθt(ω + ω′)]

×[|r(ω) + r∗(ω′)|2 + |s(ω) − s∗(ω′)|2] + O(ǫ2)


Energy conservation is fulfilled: the dynamical energy at anytime t equals, with the opposite sign, the work performed bythe reaction force up to that time t

〈E(t)〉 = −ǫ

∫t

0

〈FHa(τ)〉g(τ)dτ




〈FHa(t)〉 = −ǫ

2π2

∫∞

0

∫∞

0

dωdω′ωω′

ω + ω′Re

[e−i(ω+ω

′)t gθt(ω + ω′)]

×[|r(ω) + r∗(ω′)|2 + |s(ω) − s∗(ω′)|2] + O(ǫ2)


Energy conservation is fulfilled: the dynamical energy at anytime t equals, with the opposite sign, the work performed bythe reaction force up to that time t

〈E(t)〉 = −ǫ

∫t

0

〈FHa(τ)〉g(τ)dτ

=⇒ Two mirrors; higher dimensions; fields of any kind


Quantum Vacuum Fluct’s & the CCThe main issue: S.A. Fulling et. al., hep-th/070209v2

energy ALWAYS gravitates, therefore the energy density of thevacuum, more precisely, the vacuum expectation value of thestress-energy tensor 〈Tµν〉 ≡ −Egµν




Appears on the rhs of Einstein’s equations:

Rµν −1

2gµνR = −8πG(Tµν − Egµν)

It affects cosmology: Tµν excitations above the vacuum





Rµν −1



Equivalent to a cosmological constant Λ = 8πGE





Rµν −1




Idea: zero point fluctuations can contribute to thecosmological constant Ya.B. Zeldovich ’68





Rµν −1




Idea: zero point fluctuations can contribute to thecosmological constant Ya.B. Zeldovich ’68

Recent observations: M. Tegmark et al. [SDSS Collab.] PRD 2004

Λ = (2.14 ± 0.13 × 10−3 eV)4 ∼ 4.32 × 10−9 erg/cm3


Vacuum Fluct & the Equival PrincipleCasimir effect 1948, same year as discovery of renormalized QED

It continues to be surrounded by controversy:




Sharp boundaries give divergences in local energy density near surface





Coupling to gravity (source in the local energy-momentum tensor)






Gravitational implications of zero-point energy:

problem due to inability to understand origin of cc or dark energy








Question: how finite Casimir energy of pair of plates couples to gravity?









Less straightforward than one might suspect!

Disparate answers: forces that depend on the orientation of the Casimirapparatus wrt gravitational field of the earth, etc











Consistency with the equivalence principle?












That is, does the renormalized Casimir energy couple to gravity

just like any other energy?














Evidence that the Vacuum Energy must be taken seriously in Gravity














Evidence that the Vacuum Energy must be taken seriously in Gravity

Boundary divergences resolved by understanding renormalization process4th Dark Side of the Universe, BUE El Cairo, 1-5 June, 2008 – p. 13/19

Casimir stress tensor between pair of parallel perfectly conducting plates, at

distance a, transverse dimensions L >> a

[Brown and Maclay, Phys. Rev. 184 (1969) 1272]

〈Tµν〉 =EC

adiag(1,−1,−1, 3)

third spatial direction is normal to plates, EC Casimir energy per unit area

EC = − π~c

720a3

Outside the plates, 〈Tµν〉 = 0





〈Tµν〉 =EC

adiag(1,−1,−1, 3)


EC = − π~c

720a3


A constant divergent term is present, both between and outside the plates,

and also in the absence of plates (no physical significance): em field respectsconformal symmetry, there is no surface divergent term (such as is present for

a minimally coupled scalar field with Dirichlet BC on plates or curved surfaces)





〈Tµν〉 =EC

adiag(1,−1,−1, 3)


EC = − π~c

720a3


A constant divergent term is present, both between and outside the plates,

and also in the absence of plates (no physical significance): em field respectsconformal symmetry, there is no surface divergent term (such as is present for

a minimally coupled scalar field with Dirichlet BC on plates or curved surfaces)

Gravitational interaction of the Casimir apparatus: use gravitational definitionof the energy-momentum tensor as variation of matter part of action:

δWm =1

2

∫ √−g δgµν T

µν (∗)

Following Schwinger (note the factor 2 in the definition),for a weak field Fulling et al define: gµν = ηµν + 2hµν


Two ways to proceed. Gauge-invariant procedure:

energy-momentum tensor of the phys sys must be conserved, so include aphysical mechanism holding the plates apart against the Casimir force

−→ Leads to complicated model-dependent calculations





Alternative: find a physically natural coordinate system, more realistic than

another. A perfect coord system is not possible in curved space, but one thatcomes closest to give distances accurately along timelike worldline is the

Fermi coord system:−→ general-relativistic extrapolation of an inertial coord frame [given by

Marzlin (1994) for a resting observer in the field of a static mass distribution]









Again, the reason one gets different answers in different coordinate systems is

that the starting point is not gauge-invariant











Bimonte et al, arXiv:hep-th/0703062: maybe Eq. (*) implicitly assumes the

equivalence principle, as a conservation law: ∇µT µν = 0













Fulling et al: Casimir energy gravitates as any other form of energy (result

obtained for a Fermi observer, relativ general of an inertial obs) cf. Feynmann















Explicit calc in Rindler coord (uniform accel obs): EC (including diverg partsrenormalizing mass of plates) has the grav mass demanded by equiv princip















Explicit calc in Rindler coord (uniform accel obs): EC (including diverg partsrenormalizing mass of plates) has the grav mass demanded by equiv princip

BUT: the renormalized total T µν must be conserved in curved s-t (gauge inv!)4th Dark Side of the Universe, BUE El Cairo, 1-5 June, 2008 – p. 15/19

CC PROBLEM

Relativistic field: collection of harmonic oscill’s (scalar field)

E0 =~ c

2

∑

n

ωn, ω = k2 + m2/~2, k = 2π/λ


CC PROBLEM


E0 =~ c

2

∑

n

ωn, ω = k2 + m2/~2, k = 2π/λ

Evaluating in a box and putting a cut-off at maximum kmax corresp’ngto QFT physics (e.g., Planck energy)

ρ ∼~ k4

Planck

16π2∼ 10123ρobs

kind of a modern (and thick!) aether R. Caldwell, S. Carroll, ...


CC PROBLEM


E0 =~ c

2

∑

n

ωn, ω = k2 + m2/~2, k = 2π/λ


ρ ∼~ k4

Planck

16π2∼ 10123ρobs


Observational tests see nothing (or very little) of it:=⇒ (new) cosmological constant problem


CC PROBLEM


E0 =~ c

2

∑

n

ωn, ω = k2 + m2/~2, k = 2π/λ


ρ ∼~ k4

Planck

16π2∼ 10123ρobs



Very difficult to solve and we do not address this question directly[Baum, Hawking, Coleman, Polchinsky, Weinberg,...]


CC PROBLEM


E0 =~ c

2

∑

n

ωn, ω = k2 + m2/~2, k = 2π/λ


ρ ∼~ k4

Planck

16π2∼ 10123ρobs



Very difficult to solve and we do not address this question directly[Baum, Hawking, Coleman, Polchinsky, Weinberg,...]

What we do consider —with relative success in some differentapproaches— is the additional contribution to the cc coming from thenon-trivial topology of space or from specific boundary conditionsimposed on braneworld models:

=⇒ kind of cosmological Casimir effect


Cosmolog Imprint of the Casimir Eff’t?Assuming one will be able to prove (in the future) that the groundvalue of the cc is zero (as many had suspected until recently), we willbe left with this incremental value coming from the topology or BCs

* L. Parker & A. Raval, VCDM, vacuum energy density* C.P. Burgess et al., hep-th/0606020 & 0510123: Susy Large ExtraDims (SLED), two 10−2mm dims, bulk vs brane Susy breaking scales* T. Padmanabhan, gr-qc/0606061: Holographic Perspective, CC is anintg const, no response of gravity to changes in bulk vac energy dens




We show (with different examples) that this value acquiresthe correct order of magnitude —corresponding to the onecoming from the observed acceleration in the expansion ofour universe— in some reasonable models involving:





(a) small and large compactified scales






(b) dS & AdS worldbranes






(b) dS & AdS worldbranes

(c) supergraviton theories (discret dims, deconstr)


Gravity Equations as Equations of StateThe cosmological constant as an “integration constant”

T. Padmanabhan; D. Blas, J. Garriga, ...




Ted Jacobson [PRL1995] obtained Einstein’s equations from

local thermodynamics arguments only






By way of generalizing black hole thermodynamics to

space-time thermodynamics as seen by a local observer








This strongly suggests, in a fundamental context:

Einstein’s Eqs to be viewed as EoS










Should, probably, not be taken as basic for quantizing gravity










Should, probably, not be taken as basic for quantizing gravity

C. Eling, R. Guedens, T. Jacobson [PRL2006]: extension to

polynomial f(R) gravity but as non-equilibrium thermodyn.

Also Erik Verlinde (personal discussions)


Jacobson’s argum non-trivially extended to f(R) gravity field

eqs, as EoS of local space-time thermodynamics

EE, P. Silva, arXiv:0804.3721v2





By means of a more general definition of local entropy, using

Wald’s definition of dynamic BH entropy

RM Wald PRD1993; V Iyer, RM Wald PRD1994








And also the concept of an effective Newton constant for

graviton exchange (effective propagator)

R. Brustein, D. Gorbonos, M. Hadad, arXiv:0712.3206











Last-days activity: S-F Wu, G-H Yang, P-M Zhang,

arXiv:0805.4044, direct extension of our results to

Brans-Dicke and scalar-tensor gravities











Last-days activity: S-F Wu, G-H Yang, P-M Zhang,

arXiv:0805.4044, direct extension of our results to

Brans-Dicke and scalar-tensor gravities

Shukran Gazillan4th Dark Side of the Universe, BUE El Cairo, 1-5 June, 2008 – p. 19/19

vacuum fluctuations in domains with moving …vacuum fluctuations and the equivalence principle ce...

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